How hot do auroras get?

They don't. Actually the colors you see which look like SOMETHING ought to be red hot or blue-hot does not come from the same physical process that makes iron bars glow red hot and white hot. What is happening is that individual atoms of nitrogen and oxygen are stimulated to emit light at only a few specific frequencies after getting kicked by charged particles raining down upon the upper atmosphere from the plasma clouds orbiting the Earth. The atoms emit the light, and it falls in parts of the optical spectrum we see as red or green.